I am wondering whether there exist a closed-form expression for the expected assortativity coefficient (http://arxiv.org/pdf/cond-mat/0205405.pdf) of an Erdos Ranyi random graph model $G(n,m)$, where $n$ is the number of nodes and $m$ is the density (http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model).
A rough approximation might be provided by the correlation of a multinomial distribution, but it doesn't seem particularly accurate when the graph density is reasonably high. In fact, the degree of a node cannot be greater than $n-1$, whereas if we regard the degree vector of $G(n,m)$ to be distributed as $Multinomial(p_1, p_2, \ldots, p_n; m)$, where $p_i = m/n$, for $i = 1, \ldots, n$, the result is that a node might have a degree greater than $n-1$ with positive probability. This is clearly wrong! In any case, if we assume the degree vector of $G(n,m)$ to be distributed as $Multinomial(p_1, \ldots, p_n; m)$, the result is that the degree correlation (which represent the assortativity coefficient) is $\frac{1}{n-1}$.
I think that many of you would probably claim that the best choice is to use a truncated multinomial, $P(d_1, \ldots, d_n | d_1 < n-1, \ldots, d_n<n-1)$, and to derive the correlation of this multivariate expression. Nonetheless, before undertaking such a tedious work I'd like to see whether someone already know a closed-form expression for the expected assortativity coefficient.
Thank you very much for your help.
